## Abstract

Let V = B(H) or S(H), where B(H) is the algebra of a bounded linear operator acting on the Hilbert space H, and S(H) is the set of selfadjoint operators in B(H). Denote the numerical range of A ∈ B(H) by W(A) = {(Ax,x): x ∈ H, (x,x) = 1}. It is shown that a surjective map φ: V → V satisfies W(AB + BA) = W(φ(A)φ(B) + φ(B)φ(A)) for all A, B ∈ V if and only if there is a unitary operator U ∈ B(H) such that φ has the form X ±U^{*}XU or X ±U^{*}X^{t} U, where X^{t} is the transpose of X with respect to a fixed orthonormal basis. In other words, the map φ or-φ is a C^{*}-isomorphism on B(H) and a Jordan isomorphism on S(H). Moreover, if H has finite dimension, then the surjective assumption on φ can be removed.

Original language | English |
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Pages (from-to) | 2907-2914 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 135 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2007 |

## Keywords

- Jordan product
- Numerical range

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